set S0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set A0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE );
set Sn = n -BitSubtracterStr x,y;
set o0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )];
deffunc H₁( set , Nat) -> Element of InnerVertices (BorrowStr (x . ($2 + 1)),(y . ($2 + 1)),$1) = BorrowOutput (x . ($2 + 1)),(y . ($2 + 1)),$1;
deffunc H₂( non empty ManySortedSign , set , Nat) -> ManySortedSign = $1 +* (BitSubtracterWithBorrowStr (x . ($3 + 1)),(y . ($3 + 1)),$2);
deffunc H₃( non empty ManySortedSign , non-empty MSAlgebra of $1, set , Nat) -> MSAlgebra of $1 +* (BitSubtracterWithBorrowStr (x . ($4 + 1)),(y . ($4 + 1)),$3) = $2 +* (BitSubtracterWithBorrowCirc (x . ($4 + 1)),(y . ($4 + 1)),$3);
A1: for S being non empty ManySortedSign
for A being non-empty MSAlgebra of S
for z being set
for n being Nat holds H₃(S,A,z,n) is non-empty MSAlgebra of H₂(S,z,n) ;
thus for A1, A2 being strict gate`2=den Boolean Circuit of n -BitSubtracterStr x,y st ex f, g, h being ManySortedSet of st
( n -BitSubtracterStr x,y = f . n & A1 = g . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra of S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = H₂(S,x,n) & g . (n + 1) = H₃(S,A,x,n) & h . (n + 1) = H₁(x,n) ) ) ) & ex f, g, h being ManySortedSet of st
( n -BitSubtracterStr x,y = f . n & A2 = g . n & f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) & h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra of S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = H₂(S,x,n) & g . (n + 1) = H₃(S,A,x,n) & h . (n + 1) = H₁(x,n) ) ) ) holds
A1 = A2 from CIRCCMB2:sch 21(A1); :: thesis: verum